PRACTICAL ENGINEERING, PROCESS, AND RELIABILITY STATISTICS.

197 Pages·2022·1.319 MB·English

by  MARK ALLEN. DURIVAGE

Practical Engineering,
Process, and
Reliability Statistics
Second Edition
Mark Allen Durivage
Milwaukee, Wisconsin
Table of Contents
List of Figures and Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Chapter 1 Point Estimates and Measures of Dispersion . . . . . . . . . . . . . . . . 1
Estimates of Central Tendency for Variables Data ...................... 1
Range for Variables Data.......................................... 4
Variance and Standard Deviation for Variables Data .................... 4
Skewness and Kurtosis for Variables Data ............................ 7
Estimates of Central Tendency for Attributes Data...................... 9
Estimates of Dispersion for Attributes Data ........................... 10
Standard Error.................................................. 11
Chapter 2 Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Confidence Interval for the Mean of Continuous Data ................... 14
Confidence Interval for the Variance and Standard Deviation ............. 15
Confidence Interval for the Fraction Nonconforming—Normal
Distribution................................................. 17
Confidence Interval for Proportion (Normal Approximation of the
Binomial Confidence Interval) .................................. 18
Small Sample Size Confidence Intervals ............................. 19
Confidence Interval for the Poisson Distributed Data ................... 20
Chapter 3 Tolerance and Prediction Intervals . . . . . . . . . . . . . . . . . . . . . . . . 21
Tolerance Intervals............................................... 21
Prediction Intervals .............................................. 22
Prediction Intervals When the Variance (σ2) Is Known................... 22
Prediction Intervals When the Variance (σ2) Is Unknown................. 24
Chapter 4 Correlation and Regression Analysis . . . . . . . . . . . . . . . . . . . . . . 27
Correlation Analysis ............................................. 27
Regression Analysis.............................................. 30
Normal Probability Plots.......................................... 31
iii
iv Table of Contents
Chapter 5 Outliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Outlier Detection Based on the Standard Deviation for a
Normal Distribution .......................................... 35
Discordance Outlier Test.......................................... 36
Outlier Detection Based on the Standard Deviation for an
Unknown Distribution......................................... 37
Outlier Detection Based on the Interquartile Range ..................... 39
Dixon’s Q Test .................................................. 39
Dean and Dixon Outlier Test....................................... 40
Grubbs’ Outlier Test ............................................. 41
Walsh’s Outlier Test.............................................. 42
Hampel’s Method for Outlier Detection .............................. 44
Chapter 6 Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Type I and Type II Errors ......................................... 47
Alpha (α) and Beta (β) Risks....................................... 47
The Effect Size Index ............................................ 48
Apportionment of Risk in Hypothesis Testing.......................... 49
The Hypothesis Test for a One-Tail (Upper-Tailed) Test . . . . . . . . . . . . . . . . . . 49
The Hypothesis Test for a One-Tail (Lower-Tailed) Test.................. 50
The Hypothesis Test for a Two-Tail Test .............................. 50
The Hypothesis Test Conclusion Statements........................... 51
Chapter 7 Sample Size Determination for Tests of Hypotheses . . . . . . . . . . 53
Sample Size Required to Test an Observed Mean versus a
Hypothesized Mean When Standard Deviation (σ) Is Known .......... 53
Sample Size Required to Test an Observed Mean versus a
Hypothesized Mean When the Standard Deviation (σ) Is Estimated
from Observed Values......................................... 55
Sample Size Required to Test for Differences in Two Observed
Means When Standard Deviation (σ) for Each Population Is Known..... 55
Sample Size Required to Test for Differences in Two Observed
Means When the Standard Deviation (σ) Is Estimated from the
Observed Data............................................... 56
Sample Size Required to Detect a Difference in the Mean
with a Given Confidence....................................... 57
Paired Sample t-Test Requirements.................................. 57
Sample Size Required for Chi-Square Test of Observed Variance to
a Hypothesized Variance....................................... 59
Sample Size Required for F-Test of Two Observed Variances ............. 61
Chapter 8 Hypothesis Testing for a Difference in Means . . . . . . . . . . . . . . . 63
Testing a Sample Mean versus a Hypothesized Mean When the
Standard Deviation (σ) Is Known ................................ 63
Table of Contents v
Testing a Sample Mean versus a Hypothesized Mean When the
Standard Deviation (σ) Is Estimated from the Sample Data............ 66
Testing for a Difference in Two Population Means—Standard
Deviations (σ) Known......................................... 68
Testing a Sample Mean versus a Hypothesized Mean When Standard
Deviation (σ) Is Estimated from the Sample Data.................... 71
Testing for a Difference in Two Population Means—Standard
Deviations (σ) Not Known and Not Assumed Equal.................. 74
Testing for Differences in Means of Paired Samples..................... 77
Hypothesis Test One Proportion .................................... 80
Testing for Differences in Two Proportions............................ 82
Testing for Differences in Count Data—Equal Sample Sizes .............. 85
Testing for Differences in Count Data—Unequal Sample Sizes ............ 87
Hypothesis Testing for Differences in Means—Confidence Interval
Approach (Standard Deviations Known)........................... 89
Hypothesis Testing for Differences in Means—Confidence Interval
Approach (Standard Deviations Not Known but Assumed Equal)....... 91
Kruskal-Wallis Means Test ........................................ 93
Chapter 9 Hypothesis Testing for a Difference in Variances . . . . . . . . . . . . 97
Testing a Variance Calculated from a Sample against a Hypothesized
Variance ................................................... 97
Testing an Observed Variance against a Hypothesized Variance—
Large Samples............................................... 99
Testing for a Difference between Two Observed Variances Using
Sample Data ................................................ 102
Testing for a Difference between Two Observed Variances Using
Large Samples............................................... 104
Levene’s Test for Equality of Variances............................... 106
Chapter 10 Discrete Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . 109
Binomial Distribution ............................................ 109
Poisson Distribution.............................................. 111
Hypergeometric Distribution....................................... 113
Geometric Distribution ........................................... 114
Negative Binomial Distribution..................................... 115
Chapter 11 Control Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Control Chart Types and Selection .................................. 117
Control Chart Interpretation ....................................... 118
X-bar and R Control Charts........................................ 121
X-bar and s Control Charts ........................................ 124
c-Charts....................................................... 128
u-Charts....................................................... 130
vi Table of Contents
np-Charts...................................................... 133
p-Charts....................................................... 135
g-Charts .................................................... 137
X and mR (Moving Range) Control Charts ........................... 140
Pre-control Charts............................................... 143
Chapter 12 Process Capability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Process Capability for Variables Data................................ 147
Process Capability Confidence Intervals.............................. 150
Process Capability for Attributes Data ............................... 152
Increasing Process Capability ...................................... 152
Fall-Out Rates .................................................. 153
Defects per Million Opportunities (DPMO) ........................... 154
Chapter 13 Acceptance Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
C = 0 Sampling Plan ............................................. 155
Average Outgoing Quality......................................... 155
Upper Risk Level (Confidence Statement)............................. 156
Sample Size Required to Find Percentage Defective with a
Given Confidence Level ....................................... 156
AQL Calculations ............................................... 157
LTPD Calculations............................................... 158
Chapter 14 ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
One-Way ANOVA............................................... 159
Two-Way ANOVA............................................... 162
Chapter 15 The Reliability Bathtub Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Chapter 16 Availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Inherent Availability ............................................. 171
Achieved Availability ............................................ 172
Operational Availability........................................... 173
Chapter 17 Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Chapter 18 Censoring and MTBF and MCBF Calculations . . . . . . . . . . . . . 177
Type I Censoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Type II Censoring ............................................... 178
Chapter 19 Confidence Intervals for MTBF/MCBF . . . . . . . . . . . . . . . . . . . 181
Testing to a Predetermined Time/Cycles.............................. 181
Testing to a Predetermined Number of Failures ........................ 182
Failure-Free Testing.............................................. 184
Chapter 20 Nonparametric and Related Test Designs . . . . . . . . . . . . . . . . . 185
Calculating Reliability in Zero-Failure Situations....................... 185
Table of Contents vii
Chapter 21 Sample Size, Reliability, and Confidence Determination . . . . . 187
Sample Size Determination Based on Confidence and
Reliability with Zero Failures Allowed............................ 187
Reliability Estimate When Sample Size Is Provided..................... 187
Sample Size Calculation with Failures Allowed ........................ 188
Reliability Estimate When Sample Sizes Are Specified.................. 189
Chapter 22 Wear-Out Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Wear-Out Distribution—Standard Deviation Known .................... 191
Wear-Out Distribution—Standard Deviation Unknown .................. 192
Wear-Out and Chance Failure Combined ............................. 192
Chapter 23 Conditional Probability of Failure . . . . . . . . . . . . . . . . . . . . . . . 195
Chapter 24 System Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Series Reliability Systems ......................................... 198
Parallel Reliability Systems........................................ 199
Combination Reliability Systems.................................... 200
Standby Parallel Systems.......................................... 201
Equal Failure Rates—Perfect Switching .............................. 202
Unequal Failure Rates—Perfect Switching ............................ 203
Equal Failure Rates—Imperfect Switching............................ 203
Unequal Failure Rates—Imperfect Switching.......................... 204
Shared Load Parallel Systems...................................... 205
k-out-of-n System................................................ 206
Bayes’ Theorem................................................. 207
Chapter 25 Stress-Strength Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Chapter 26 Weibull Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Weibull Distribution Parameters .................................... 213
Weibull Fit Determination......................................... 214
Chapter 27 Log-Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Chapter 28 Beta Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Chapter 29 Taguchi Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Nominal is Best S/N Ratio......................................... 223
Larger is Better S/N Ratio......................................... 223
Smaller is Better S/N Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
Zero is Best (Signed Value) S/N Ratio................................ 224
Fraction Defective S/N Ratio....................................... 225
Ordered Categorical S/N Ratio ..................................... 225
Convert dB to Percentage ......................................... 225
viii Table of Contents
Appendix A Normal Distribution Probability Points—Area below Z . . . . . 227
Appendix B Normal Distribution Probability Points—Area above Z . . . . . 229
Appendix C Selected Single-Sided Normal Distribution
Probability Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Appendix D Selected Double-Sided Normal Distribution
Probability Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Appendix E Percentage Points of the Student’s t-Distribution . . . . . . . . . . . 235
Appendix F Distribution of the Chi-Square . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Appendix G Percentages of the F-Distribution . . . . . . . . . . . . . . . . . . . . . . . 239
Appendix H Tolerance Interval Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
Appendix I Critical Values of the Correlation Coeffecient . . . . . . . . . . . . . 257
Appendix J Critical Values of the Dean and Dixon Outlier Test . . . . . . . . . 259
Appendix K Critical Values for the Grubbs’ Outlier Test . . . . . . . . . . . . . . 261
Appendix L Critical Values for the Discordance Outlier Test . . . . . . . . . . . 263
Appendix M The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
Appendix N The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Appendix O Control Chart Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
Appendix P C = 0 Sampling Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Appendix Q Fall-Out Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
Appendix R Beta Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
Appendix S Gamma Function of Γ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
Appendix T Selected Median Rank Percentages . . . . . . . . . . . . . . . . . . . . . . 307
Appendix U Weibull Graph Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
American Society for Quality, Quality Press, Milwaukee, WI 53203
© 2015, 2022 by ASQ
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Printed in the United States of America.
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Library of Congress Cataloging-in-Publication data
Names: Durivage, Mark Allen, author.
Title: Practical engineering, process, and reliability statistics, second edition /
By Mark Allen Durivage.
Description: Includes bibliographical references and index. | Milwaukee, WI:
ASQ Quality Press, 2022.
Identifiers: LCCN: 2022933438 | ISBN 978-1-63694-015-1 (paperback) |
978-1-63694-016-8 (epub)
Subjects: LCSH Engineering—Statistical methods. | Reliability (Engineering)—
Statistical methods. | BISAC MATHEMATICS / Probability & Statistics /
General | TECHNOLOGY & ENGINEERING / Engineering / General
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Acknowledgments
would like to acknowledge the previous work of Robert A. Dovich in Quality
I
Engineering Statistics and Reliability Statistics. This book is an expansion of his efforts
in an attempt to continue the method of presenting statistical applications in a simple,
easy-to-follow format. Several sections of this book come directly from his previous work.
I have made some changes to clarify and augment some of his points and present the topics
in a consistent manner. I would like to thank those who have inspired, taught, and trained
me throughout my academic and professional career. Lastly, I would like to acknowledge
the patience of my wife, Dawn, and my sons, Jack and Sam, which allowed me time to
research, write, and update Practical Engineering, Process, and Reliability Statistics,
second edition.
xv
List of Figures and Tables
Figure 1.1 Normal distribution ............................................ 3
Figure 1.2 Negatively skewed distribution ................................... 3
Figure 1.3 Positively skewed distribution .................................... 3
Figure 1.4 Skewness .................................................... 8
Figure 1.5 Kurtosis ..................................................... 8
Figure 1.6 Sample size and the standard error relationship ...................... 12
Figure 2.1 Relationship between the confidence interval and sample size
(confidence level constant) ...................................... 13
Figure 2.2 Relationship between the confidence interval and confidence level
(sample size constant) .......................................... 14
Figure 4.1 Relative degrees of correlation ................................... 28
Table 4.1 Correlation axis table .......................................... 29
Table 4.2 Example calculation summary ................................... 29
Figure 4.2 Nonlinear quadratic and cubic relationships ......................... 31
Figure 4.3 Right-skewed distribution ....................................... 32
Figure 4.4 Left-skewed distribution ........................................ 32
Figure 4.5 Short-tailed distribution ........................................ 32
Figure 4.6 Long-tailed distribution ......................................... 32
Figure 4.7 Normal probability plot for strength ............................... 34
Table 4.3 Calculation summary .......................................... 34
Figure 5.1 Normal distribution ............................................ 36
Table 5.1 Selected k values .............................................. 37
Figure 5.2 Unknown distribution .......................................... 38
Table 5.2 Selected critical Q values ....................................... 40
Table 6.1 Hypothesis truth table .......................................... 47
Figure 6.1 Representation of a one-tail (upper-tailed) test ....................... 50
Figure 6.2 Representation of a one-tail (lower-tailed) test ....................... 50
Figure 6.3 Representation of a two-tail test .................................. 51
Table 7.1 Hypothesis constant table for two-tail and one-tail tests ............... 55
ix